Concept: Solving Inequalities Name:

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Hilary McBride
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1 Concept: Solving Inequalities Name: You should have completed Equations Section 7 Part A: Solving Inequalities before beginning this handout. COMPUTER COMPONENT Instructions: In follow the Content Menu path: Equations > Solving Inequalities NOTE: Use the Menu button in order to get to the lesson where you left off. Work through all Sub Lessons of the following Lessons in order: Graphing Linear inequalities in Two Variables Solving Systems of Linear Inequalities by Graphing Linear Programming Additional Required Materials: Pencil Crayons As you work through the computer exercises, you will be prompted to make notes in your notebook/math journal. NOTES: Graphing Linear Inequalities in Two Variables The solution is not a on the graph. The solution is a of the graph. The graph displays the linear inequality y < x + 1 The points on the boundary line ( line) represent all points where y x+1. The points below the boundary line ( line) represent all points where y x

The graph displays the linear inequality y -2 x - 1 The points on the boundary line ( line) represent all points where y -2 x - 1 The points above the boundary line ( line) represent all points where

2 The graph displays the linear inequality y -2 x - 1 The points on the boundary line ( line) represent all points where y -2 x - 1 The points above the boundary line ( line) represent all points where y -2 x - 1. Checking a point on either side of the boundary line will help determine if that region is in the solution. Steps to graphing Linear Inequalities in Two Variables: (Fill in the blanks) Step 1: the boundary line. Step 2: Determine if the boundary line is or. Step 3: Pick a point on either side of the line and check if that point is in the solution. Example: y < x + 1 Check (0, 0) y < x + 1 < + 1 < The graph of the linear inequality y < 2x - 1 is shown. If it is true then (0, 0) is part of the solution (a) Color the region covering points that make the inequality true. (b) Indicate the boundary line. (c) Write the equation for the boundary line. 2

3 The graph of the linear inequality y -x - 2 is shown. (a) Color the region covering points that make the inequality true. (b) Indicate the boundary line. (c) Write the equation for the boundary line. Explain why some boundary lines are dotted and others are solid. Solving Systems of Linear Inequalities by Graphing Graph each inequality. The solution to the system will be the where the shadings from each inequality one another. 3

4 Linear Programming: The food processing industry is perhaps the second most active user of linear programming, where it was first used to determine shipping of ketchup from a few plants to many warehouse. books.google.ca/books?isbn= Linear Programming is a method used to manage and. Inequalities are used as that the business. The of the of constraints is called the. The of the region are possible and values. In the following graph indicate the following: (a) Feasible region (b) Vertices of the feasible region 4

5 OFF COMPUTER EXERCISES 1. Graph the following inequalities. (a) y > 2 x - 4 (b) y < -3 x

6 (a) y x Solve each system of inequalities. (Hint: Find the feasible region.) (a) y 2 x - 4 y < - x

7 (b) y > x - 3 y 2 x - 5 (c) y > 1 y < x

8 3. A tire manufacturer makes two types of tires; regular and winter. A maximum of 200 regular tires and 125 winter tires can be manufactured in one day. The finishing machine can only handle up to 250 tires in one day. The profit on each regular tire is $25 and on each winter tire is $30. How many of each type should be made in order to maximize profits? Solution: Let the number of regular tires be represented by x and the number of winter tires be represented by y. We must now determine our constraints. x y machine) x + y (Hint: The finishing Graph each of the three constraints above. Don't forget to shade the appropriate areas in order to find the feasible region. You should find that your feasible region is formed by 5 vertices: (0, 0), (0, ), (, 0), (,50), (125, ) 8

9 Remembering that our goal is to find what combination will produce the maximum profit, we now want to substitute each set of vertices into the profit equation P = 25x + 30y (This information was given to us in the question) After all 5 substitutions have been made, what combination of tire can you conclude will bring the company maximum profit? Justify your answer. 9

Concept: Solving Inequalities Name:

Concept: Solving Inequalities Name: You should have completed Equations Section 7 Part A: Solving Inequalities before beginning this handout. COMPUTER COMPONENT Instructions: In follow the Content Menu